Question

find the product. (simplify your answer completely.) 3b(4b - 3)² \boxed{24b² - 18b} (with a red cross mark on the boxed expression)

Explanation:

Step1: Expand the square term

First, we need to expand \((4b - 3)^2\) using the formula \((a - b)^2=a^2-2ab + b^2\), where \(a = 4b\) and \(b=3\).
\((4b - 3)^2=(4b)^2-2\times(4b)\times3+3^2=16b^2-24b + 9\)

Step2: Multiply by \(3b\)

Now, we multiply \(3b\) with the expanded form \(16b^2-24b + 9\) using the distributive property \(c(a + b + d)=ca+cb+cd\), where \(c = 3b\), \(a=16b^2\), \(b=- 24b\) and \(d = 9\).
\(3b\times(16b^2-24b + 9)=3b\times16b^2-3b\times24b+3b\times9\)
\(=48b^{3}-72b^{2}+27b\)

Wait, but the original problem seems to have a cross - out on \(24b^{2}-18b\), maybe there was a mistake in the initial problem understanding. Let's re - examine the problem. If the problem is to simplify \(3b(4b - 3)^2\) (ignoring the crossed - out part as it might be a wrong option or a distractor):

Step1: Expand \((4b - 3)^2\)

Using \((x - y)^2=x^{2}-2xy + y^{2}\), for \(x = 4b\) and \(y = 3\), we have \((4b-3)^2=(4b)^2-2\times4b\times3 + 3^2=16b^{2}-24b + 9\)

Step2: Distribute \(3b\)

\(3b\times(16b^{2}-24b + 9)=3b\times16b^{2}-3b\times24b+3b\times9\)
\(=48b^{3}-72b^{2}+27b\)

If the problem was supposed to be \(3b(4b - 3)\) (maybe a typo in the square), then:

Step1: Distribute \(3b\)

Using the distributive property \(a(b - c)=ab-ac\), where \(a = 3b\), \(b = 4b\) and \(c = 3\)
\(3b(4b-3)=3b\times4b-3b\times3\)
\(=12b^{2}-9b\)

But from the crossed - out term \(24b^{2}-18b = 6b(4b - 3)\), and if we consider the correct expansion of \(3b(4b - 3)^2\) as above. However, if we assume that the problem was \(6b(4b - 3)\) (since \(24b^{2}-18b=6b(4b - 3)\)):

Step1: Distribute \(6b\)

\(6b(4b - 3)=6b\times4b-6b\times3\)
\(=24b^{2}-18b\)

Ah, maybe the problem was to show that \(6b(4b - 3)=24b^{2}-18b\) and then there was a mistake in the exponent. If we go back to the original problem statement "Find the product. (Simplify your answer completely.) \(3b(4b - 3)^2\)" (ignoring the crossed - out part):

The correct expansion is:

Step1: Expand the square

\((4b - 3)^2=16b^{2}-24b + 9\)

Step2: Multiply by \(3b\)

\(3b\times(16b^{2}-24b + 9)=3b\times16b^{2}-3b\times24b+3b\times9\)
\(=48b^{3}-72b^{2}+27b\)

But if the problem was a simple multiplication like \(6b(4b - 3)\) (matching the crossed - out term):

Step1: Apply distributive property

\(6b(4b-3)=6b\times4b-6b\times3\)
\(=24b^{2}-18b\)

Answer:

If the problem is \(3b(4b - 3)^2\), the answer is \(48b^{3}-72b^{2}+27b\); if the problem is \(6b(4b - 3)\) (matching the crossed - out term), the answer is \(24b^{2}-18b\)